Overview

This document provides an overview of the univariate relationships of all covariates with the absolute parameter deviation. We separate the relationships by focussing on one method as the target method and then investigating the relationships for each of the remaining methods with this method.

DV: Absolute Deviation from Complete Pooling MLE Estimate

We begin by investigating the abolsute relationship from the simplest method, the complete pooling MLE method (i.e., y always refers to Comp MLE" and x refers to the other method in the pair). This leaves us with observations for the analysis.

Effect of Method

We can also look at the histogram of the absolute deviation across methods.

Effects of Continuous Covariates

In the following plots, the blue line shows the fitted model (in case it is not a simple linear relationship, the transformation of the independent variable is given in parentheses in the x-axis label). The \(R^2\) value shon in the plot is the \(R^2\) of this model (i.e., the blue line). The red line shows a GAM on the independent variable with shrinkage applied thin plagte regression spline.

In case observations had to be removed for the analysis, the percentage of removed (rem) observations is also shown in the x-axis caption.

Effect of Parameter Estimate

Standard Error

Hetereogeneity

The data suggests a step-like relationship such that only values that are at or near zero show a considerable probability of non-zero absolute deviations. To look at this further, we can see how probable it is to observe values near zero. The following table shows that at least 80% of observations have a log1p value that is very near to zero.

## # A tibble: 8 x 3
##   cond_x     less_than_00001 less_than_01
##   <fct>                <dbl>        <dbl>
## 1 Comp Bayes           0.819        0.905
## 2 No asy               0.823        0.914
## 3 No PB                0.822        0.915
## 4 No NPB               0.832        0.925
## 5 No Bayes             0.812        0.908
## 6 Beta PP              0.814        0.909
## 7 Trait_u PP           0.873        0.914
## 8 Trait PP             0.816        0.904

If we look at the conditionmal disttirbution of absolute deviation whether or not it is very near to zero, we can see that there is some evidence for the step-like relationship, but the pattern is not overwhelming.

Rho

Fungibility

Model Fit

Relative Parameter Information

The reason both plots look pretty much the same is that both relative parameter information variables are highly correlated, \(r \approx 1\). We therefore focus on one of the two below (rel_par_weight_y).

Using a logarithm:

And removing all with a relative weight of roughly 1:

Relative Parameter N

We can also consider the relative N. As tehse are again highly correlated (\(r \approx 1\)), we use y again exclusivly:

Here it makes sense to trim the x-axis a bit: